Statistical aspects of ultracold resonant scattering

Compared to purely atomic collisions, ultracold collisions involving molecules have the potential to support a much larger number of Fano-Feshbach resonances due to the huge amount of ro-vibrational states available. In order to handle such ultracold atom-molecule collisions, we formulate a theory that incorporates the ro-vibrational Fano-Feshbach resonances in a statistical manner while treating the physics of the long-range scattering, which is sensitive to such things as hyperfine states, collision energy, and any applied electromagnetic fields, exactly within multichannel quantum defect theory. Uniting these two techniques, we can assess the influence of highly resonant scattering in the threshold regime, and in particular its dependence on the hyperfine state selected for the collision. This allows us to explore the onset of Ericson fluctuations in the regime of overlapping resonances, which are well known in nuclear physics but completely unexplored in the ultracold domain.

Figure 1

Schematics of our MQDT approach (not to scale). In the long-range part, $R>R_m$, we only consider the ro-vibrational ground state of the molecule, but include all $N_a$ atomic and molecular hyperfine states and different partial waves (not shown). In the asymptotic region, $R\to \infty$, $N_o$ of them are energetically open and $N_c$ are closed. The MQDT treatment transforms the short-range $K$ matrix ${\underline{K}}^{\mathrm{sr}}$, defined at $R_m$, into a physical scattering matrix ${\underline{S}}^{\mathrm{phys}}$ from which quantities such as elastic and inelastic cross sections can be deduced. ${\underline{K}}^{\mathrm{sr}}$ includes the information on the ro-vibrational resonances which occur in the short-range part, $R<R_m$.

Figure 2

Schematic overview (not to scale) of the origin of the short-range resonances and their distribution. (Bottom) The atom-molecule potential is modeled by a Lennard-Jones potential, Eq. (19). The resonant channels stem from ro-vibrationally excited states of the molecule. For ultracold temperatures, the incident and outgoing scattering channels are restricted to the ground ro-vibrational state of the molecule. Within this ground state, the various spin states are treated explicitly by means of MQDT. (Top) Exemplary short-range spectrum for $s$-wave collisions of Rb+KRb; it is constructed to satisfy the Wigner-Dyson distribution for the nearest neighbors [cf. Eq. (15)].

Figure 3

Elastic partial wave cross sections for K+LiK collisions; the top panel shows the corresponding cumulative elastic cross sections. The incident channel is the absolute ground state; the maximum collision energy is twice the van der Waals energy scale $E_{\mathrm{vdW}}$.

Figure 4

Elastic cross section for Rb+KRb collisions in the absolute ground state. The maximum collision energy is twice the van der Waals energy scale $E_{\mathrm{vdW}}$. (a) Shows the sum of the individual partial waves depicted in (b). (c) Provides a comparison of higher ($L\ge 2$) partial wave cross sections including [black lines, same as in (b)] and omitting [orange(gray) lines] the long-range phase shift $\tan \delta \propto k^4$ due to the $C_6/R^6$ van der Waals dispersion potential.

Figure 5

Elastic $s$-wave rate constants for Rb+KRb collisions in the absolute ground state. Depicted is the thermalized rate constant for a temperature of 1 $\mu$K, 10 $\mu$K, and 100 $\mu$K, respectively (top to bottom).

Figure 6

(a) Onset of Ericson fluctuations in ultracold Rb+KRb $s$-wave collisions at a collision energy of 100 nK. Shown is the mean resonance width (dots and solid line) of elastic cross sections as a function of the number of open channels. See text for further details. (Insets) Exemplary elastic cross sections for $N_o=2$ and $N_o=9$ open channels, respectively. (b) Atomic Zeeman shift of the scattering channels. The molecular nuclear spin gives rise to an additional magnetic field dependence which splits each atomic line into a number of sublevels; on the given scale, these sublevels are not visible. The channel indices equal the number of open channels $N_o$.

Figure 7

Influence of the Wigner-Dyson distribution on the correlation function Eq. (27). Shown is the calculated width ${\Gamma }_{\mathrm{out}}$ of the correlation function as a function of the input width ${\Gamma }_{\mathrm{in}}$ for the Lorentzian model spectrum Eq. (32). (Orange dotted line) Resonances are distributed according to the Wigner-Dyson distribution Eq. (15). (Green solid line) The same resonances are normally distributed. The shaded areas indicate the standard deviation for the given sample of 30 individual runs.